AIDJEX Bulletin #4: Water Stress Studies

AIDJEX BULLETIN No. 4 January 1971 WATER STRESS STUDIES Arctic Ice Dynamics Joint Experiment Joseph 0. Fletcher, Program Coordinator Division of Marine Resources University of Washington Seattle, Washington 98105 UNIVERSITY OF WASHINGTON Division of Marine Resources The AIDJEX Bulletin aims to provide both a fomun for discussing AIDJEX issues and a soume of infomation pertinent to all AIDJEX participants. The Bulletin series will be numbered and dated for easy reference and subtitled according to the contents of each issue. A status report will appear periodically as an issue. Other issues will contain technical material c2osely related to AIDJEX, informa2 reports on theoretical and. field work, trans2ations of relevant scientific reports, and discussion of interim AIDJEX results or prJoblems. Ycu are encouraged to send your comments and contribu- tions to Mrs. Alma Johnson AIDJEX Bul Zetin 371.0 Brooklyn Avenue N. E. Seattle, Washington 981 05 ii AIDJEX BULLETIN No. 4 January 1971 Water Stress Studies TABLE OF CONTENTS AIDJEX OCEANOGRAPHIC INVESTIGATIONS --. J. Dungan Smith A REPORT ON THE 1970 AIDJEX PILOT STUDY --. L. K. Coachman and J. Dungan Smith AN ARCTIC UNDER-ICE DIVING EXPERIMENT 39 --. Patrick Martin 1971 AIDJEX WATER STRESS PILOT STUDIES INTRODUCTION 42 -- J. Dungan Smith LAMONT MEASUREMENTS OF WATER STRESS AND OCEAN CURRENTS 44 -- Kenneth Hunkins UNIVERSITY OF WASHINGTON WATER STRESS STUDIES 48 -- L. K. Coachman and J. Dungan Smith REFERENCES 54 iii AIDJEX OCEANOGRAPHIC INVESTIGATIONS 3. Dungan Smith Departments of Oceanography and Geophysics University of Washington Seattle, Washington I. INTRODUCTION During the main phase of AIDJEX, scheduled for 1973, the air stress on the ice surface and the motion of the ice will be monitored. The oceanographic measurements will seek to determine the stress on the underside of the ice from the roughness of the underside and the relative currents. The ice roughness has a variety of scales ranging from a few meters under old floes to several tens of meters under pressure ridges, and slopes of 1:20 or steeper are common even under old floes. (For example, see Figure 3 in the 1970 Pilot Study Report in this Bulletin.) The motion of both ice and water involves the solution of complicated, time-dependent problems. The flow between the bottom of the sea ice and the quasi-geostrophic currents in the upper Arctic Ocean can be divided into two layers: a frictional boundary layer extending from the base of the ice to a distance of a few meters below the ice sheet, and an Ekman boundary layer extending from a few meters below the ice sheet to several tens of meters below the ice. In the frictional boundary layer, the turbulence is generated locally by the shear in the velocity field (which in turn is due to the presence of the boundary), and the Coriolis effect is negligible. The turbulent mixing in this region is proportional to the local shear velocity and the distance from the boundary. In the Ekman layer, the Coriolis effect is of prime importance, and the turbulent mixing is more or less independent of distance from the boundary as shown by Hunkins (1966). Preliminary analysis 1 of Smith's measurements described in the 1970 ATDJEX Pilot Study Report (this BuILletin) also indicates a constant turbulent mixing coefficient or eddy viscosity in the Ekman layer. In the atmosphere, the separation of the planetary boundary layer into two regions is also possible, The frictional boundary layer of the atmosphere is several tens of meters thick, and the Ekman layer is several hundred meters thick. The topography over much of the ice surface is several tens of centimeters in height, and even most pressure ridges are small relative to the thickness of the frictional boundary layer. This means that the Ekman layer is generally isolated from the surface topography, which, except near pressure ridges, can be considered as a surface roughness in an otherwise uniform flow. Moreover, during most periods of interest to the ice deformation problem, the wind speed at the top of the frictional boundary layer is high relative to the drift speed of the ice, permitting the ice to be considered quasi-steady for stress calculations. The situation is quite different in the oceanic boundary layer. The under-ice topography as mentioned previously is usually comparable in height to the frictional boundary layer and the pressure ridges are often comparable to ths thickness of the Ekman layer. This results in a nonuniform Ekman layer as well as a nonuniform frictional boundary layer under much of the ice. In addition to this complication, the drift speed of the ice is often comparable to the flow speed in the Ekman layer. This tends to make the oceanic boundary layer unsteady as well as nonuniform. These complications may necessitate an approach to under-ice boundary layer studies somewhat different from that used in atmospheric boundary layer studies--for example, a greater dependence on theoretical calculations. In any case, they require a careful examination of the nature of the oceanic boundary layer and its relationship to both the ice drift and the flow in the upper Arctic Ocean. For this reason, a fairly extensive oceanographic program was planned for the 1970 AIDJEX Pilot Study, which was carried out by the University of Washington in conjunction with the Canadian Polar Contintental Shelf Project at their ice floe camp in March (Camp 200). An even more extensive oceanographic program is planned for March 1971 at Camp 200. These studies are described in this Bulletin. 2 11, THEORETICAL CONSIDERATIONS If the frictional boundary layer is steady and uniform in J: and y, then a simple similarity argument in which the variables u, z, and au/az are assumed to define the problem leads to the expression U u=$~nZ zO Here u is the velocity a distance z from the boundary; U* = dsis the square root of the boundary shear stress (Tb) divided by the square root of the fluid density (p) and is called the shear velocity or friction velocity; k is a constant of proportionality called von Karman's constant and found experimentally to be about 0.40; and zo is a constant of integra- tion related to the roughness of the boundary. If k, is an appropriate measure of the local roughness of the boundary and if the "roughness Reynolds number" R, = u,k,/v (where L, is the kinematic viscosity of the fluid) is less than 3, then the boundary is called "hydraulically smooth" and Zo = V/9u, (see Schlichting, 1960, p. 519ff). On the other hand, if R, is large enough, z0 a k, and the boundary is called "hydraulically rough.'' For randomly distributed sand grains, "hydraulically rough flow" occurs at R, > 100; in this case, zo = k,/30 (see Schlichting, 1960, p. 519ff .) . For wave-type roughness, hydraulically rough flow occurs at considerably higher roughness Reynolds numbers. What is considered roughness by some writers is considered to be nonuniformity in the flow by other writers; but if the boundary is uneven with wavelengths on the order of the thickness of the frictional boundary layer or larger, it is best to consider the flow nonuniform rather than to try to treat the topography as roughness elements. By this criterion the flow in much of the frictional boundary layer under the polar ice is nonuniform, Nonuniform two-dimensional frictional boundary layers have been considered by several writers concerned with the dynamics of deformable boundaries, Much of the recent work on air-sea interaction has been of this type (e.g., Miles, 1967); however, in this case the problem is complicated by the motion of the boundary and by the fact that the boundary is considered to be thick relative to the wavelengths of the surface waves. 3 On the other hand, Smith (1970) in studying the stability of a sand bed subjected to a shear flow, has considered the boundary to be quasi-steady and has therefore provided a solution to flow over a fixed boundary of small-amplitude topography. Unfortunately, this theory uses a constant eddy viscosity which is somewhat unrealistic for natural frictional boundary layers. Although it is presently being expanded to include the case of a linearly varying eddy viscosity, results of these calculations are not yet available. Smith's theory, like that of Miles, is a first-order theory and therefore applies only to small-amplitude topography; nevertheless, it has been used with some success in predicting the boundary shear-stress distri- bution over finite-amplitude Columbia River sand waves with wavelengths of about 60 meters and heights of about 3 meters (Smith, 1969). The theory is compared to Preston tube measurements of boundary shear stress in the abovementioned report and is shown to reproduce the major features of the measurements with approximately the correct amplitudes. Further unpublished work in this area indicates that while the first- order, constant eddy viscosity theory reproduces the major structure of the boundary shear-stress profile, it overestimates the accelerative effects and underestimates the decelerative effects. This agrees with flume measurements made under Smith's direction by Reeder (1970), which show that second- order effects are about 25% as important as first-order effects in regard to flow over sinusoidal boundaries with height to wavelength ratios of 1:15, the latter being a rather common harmonic in natural topography including the Columbia River sand waves and the underside of the arctic ice. No second- order thleory is now available, and no three-dimensional theory has yet been attempted. The boundary shear stress calculated from this theory, called "skin friction" by aerodynamicists , does not include the "form drag" or "pressure drag" due to the boundary geometry.

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